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Chapter 9. Rotation of Rigid Bodies. Goals for Chapter 9. To describe rotation in terms of: angular coordinates ( q ) angular velocity ( w ) angular acceleration ( a ) To analyze rotation with constant angular acceleration

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Chapter 9

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Chapter 9
Chapter 9

Rotation of Rigid Bodies


Goals for chapter 9
Goals for Chapter 9

  • To describe rotation in terms of:

    • angular coordinates (q)

    • angular velocity (w)

    • angular acceleration (a)

  • To analyze rotation with constant angular acceleration

  • To relate rotation to the linear velocity and linear acceleration of a point on a body


  • Goals for chapter 91
    Goals for Chapter 9

    • To understand moment of inertia (I):

      • how it depends upon rotation axes

      • how it relates to rotational kinetic energy

      • how it is calculated


    Introduction
    Introduction

    • A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects.

    • Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid.


    Angular coordinate
    Angular coordinate

    • Consider a meter with a needle rotating about a fixed axis.

    • Angle  (in radians) that needle makes with +x-axis is a coordinate for rotation.


    Angular coordinates
    Angular coordinates

    • Example: A car’s speedometer needle rotates about a fixed axis.

    • Angle  the needle makes with negativex-axis is the coordinate for rotation.

    • KEY: Define your directions!

    55

    45

    35

    25

    15

    q


    Units of angles
    Units of angles

    • Angles in radians  = s/r.

    • One complete revolution is 360° = 2π radians.


    Angular velocity
    Angular velocity

    • Angular displacement of a body is  = 2 – 1 (in radians)

    • Average angular velocityof a body is av = /t(in radians/second)


    Angular velocity1
    Angular velocity

    • The rotation axis matters!

    • Subscript z means that the rotation is about the z-axis.

    • Instantaneous angular velocity is z = d/dt.

    • Counterclockwise rotation is positive;

    • Clockwise rotation is negative.


    Calculating angular velocity
    Calculating angular velocity

    • Flywheel diameter 0.36 m;

    • Suppose q(t) = (2.0 rad/s3) t3


    Calculating angular velocity1
    Calculating angular velocity

    • Flywheel diameter 0.36 m; q = (2.0 rad/s3) t3

      • Find q at t1 = 2.0 s and t2 = 5.0 s

      • Find distance rim moves in that interval

      • Find average angular velocity in rad/sec & rev/min

      • Find instantaneous angular velocities at t1 & t2


    Angular velocity is a vector
    Angular velocity is a vector

    • Angular velocity is defined as a vector whose direction is given by the right-hand rule.


    Angular acceleration
    Angular acceleration

    • Average angular acceleration is avg-z = z/t.

    • Instantaneous angular acceleration is z = dz/dt = d2/dt2.


    Angular acceleration1
    Angular acceleration

    • Average angular acceleration is avg-z = z/t.

    • Instantaneous angular acceleration is z = dz/dt = d2/dt2.

    For same flywheel with dia = 0.36 m; q(t) = (2.0 rad/s3) t3

    find average angular accelerations between t1 & t2, & instantaneous accelerations at those t’s


    Angular acceleration as a vector
    Angular acceleration as a vector

    • For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.


    Angular acceleration as a vector1
    Angular acceleration as a vector

    • For a fixedrotation axis, angular acceleration a and angular velocity w vectors both lie along that axis.

    • BUT THEY DON’T HAVE TO BE IN THE SAME DIRECTION!

    w Speeds up!

    w Slows down!


    Rotation with constant angular acceleration
    Rotation with constant angular acceleration

    • Linear and Angular Motion with constant acceleration equations are very similar!


    Rotation of a blu ray disc
    Rotation of a Blu-ray disc

    • A Blu-ray disc is coming to rest after being played.

    • @ t = 0, w = 27.5 rad/sec; a = -10.0 rad/s2

    • What is w at t = 0.3 seconds?

    • What angle does PQ make with x axis then?


    Relating linear and angular kinematics
    Relating linear and angular kinematics

    • For a point a distance r from the axis of rotation:

      its linear speed is v = r (meters/sec)


    Relating linear and angular kinematics1
    Relating linear and angular kinematics

    • For a point a distance r from the axis of rotation:

      its linear tangential acceleration is atan = r (m/s2)

      its centripetal (radial) acceleration is arad = v2/r = r


    An athlete throwing a discus
    An athlete throwing a discus

    • Whirl discus in circle of r = 80 cm; at some time t athlete is rotating at 10.0 rad/sec; speed increasing at 50.0 rad/sec/sec.

    • Find tangential and centripetal accelerations and overall magnitude of acceleration


    Designing a propeller
    Designing a propeller

    • Say rotation of propeller is at a constant 2400 rpm, as plane flies forward at 75.0 m/s at constant speed.

    • But…tips of propellers must move slower than 270 m/s to avoid excessive noise.

    • What is maximum propeller radius?

    • What is acceleration of the tip?


    Designing a propeller1
    Designing a propeller

    • Tips of propellers must move slower than than 270 m/s to avoid excessive noise. What is maximum propeller radius? What is acceleration of the tip?


    Moments of inertia
    Moments of inertia

    How much force it takes to get something rotating, and how much energy it has when rotating, depends on WHERE the mass is in relation to the rotation axis.


    Moments of inertia1
    Moments of inertia

    • Getting MORE mass, FARTHER from the axis, to rotate will take more force!

    • Some rotating at the same rate with more mass farther away will have more KE!





    Rotational kinetic energy
    Rotational kinetic energy

    • The moment of inertia of a set of discrete particles is

    • I = m1r12 + m2r22 + … = miri2

    • Rotational kinetic energy of rigid body with moment of inertia IKE(rotation) = 1/2 I2 (still in Joules!)

    • Since I varies by location, KE varies depending upon axis!


    Rotational kinetic energy example 9 6
    Rotational kinetic energy example 9.6

    • What is I about A?

    • What is I about B/C?

    • What is KE if it rotates through A with w=4.0 rad/sec?


    An unwinding cable
    An unwinding cable

    • Wrap a light, non-stretching cable around solid cylinder of mass 50 kg; diameter .120 m. Pull for 2.0 m with constant force of 9.0 N. What is final angular speed and final linear speed of cable?


    More on an unwinding cable
    More on an unwinding cable

    Consider falling mass “m” tied to rotating wheel of mass Mand radius R

    What is the resulting speed of the small mass when it reaches the bottom?


    More on an unwinding cable1
    More on an unwinding cable

    Consider falling mass “m” tied to rotating wheel of mass Mand radius R

    What is the resulting speed of the small mass when it reaches the bottom?

    Method 1: Energy!


    More on an unwinding cable2
    More on an unwinding cable

    mgh

    ½ mv2+ ½ Iw2


    More on an unwinding cable3
    More on an unwinding cable

    mgh = ½ mv2+ ½ Iw2

    I = ½ MR2 for disk

    mgh = ½ (m+ ½ M) v2

    V = [2gh/(1+M/2m)] ½


    The parallel axis theorem
    The parallel-axis theorem

    • What happens if you rotate about an EXTERNAL axis, not internal?

      • Spinning planet orbiting around the Sun

      • Rotating ball bearing orbiting in the bearing

      • Mass on turntable


    The parallel axis theorem1
    The parallel-axis theorem

    • What happens if you rotate about an EXTERNAL axis, not internal?

    • Effect is a COMBINATION of TWO rotations

      • Object itself spinning;

      • Point mass orbiting

    • Net rotational inertia combines both:

    • The parallel-axis theorem is: IP = Icm + Md2


    The parallel axis theorem2
    The parallel-axis theorem

    • The parallel-axis theorem is: IP = Icm + Md2.

    M

    d


    The parallel axis theorem3
    The parallel-axis theorem

    • The parallel-axis theorem is: IP = Icm+Md2.

    • Mass 3.6 kg, I = 0.132 kg-m2 through P. What is I about parallel axis through center of mass?


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